In grade school, you probably learned the rule that an integer is divisible by if and only if the number of it's digits is divisible by . There is a similar criteria for divisibility by which sometimes goes by the name ``casting out nines''.
From the m-ary expansion of an integer, one can deduce other useful divisibility criteria. In this section, we outline a few of the better-known divisibility tests based on the decimal expansion of an integer, and defer the proofs of these tests to section 1.7 (although the reader is invited to prove as many of these tests as she/he can).
Let
where are the digits.For example, .
For a proof of this criterion, see Example 1.7.6.
For a proof of this criterion, see Example 1.7.6.
For example, .
For example, .
In fact, this divisibility rule is the basis for the ``ISBN code'' (an error-detecting code used internationally in labeling books) whch we shall study later.
For example, .